Impedance transformer



May 3, 1'938. E. NORTON ET Al.

IMPEDANCE TRNSFORMER Filed Sept. 30, 1936 2 Sheets-Sheet 1 ATTORA/Eff Patented May 3, 1938 v UNITED STATES PATENT OFFICE IMPEDANCE TRANSFORMER Edward L. Norton, Summit, N. J., and Ronald F.

yWick, St. George, N. Y., assignors to Bell Telephone Laboratories, Incorporated, New York, N. Y., a. corporation of New York Application September 30, 1936, Serial No. 103,316

' 1s claims. (o1. 17a-44) 'I'his invention relates to impedance transformrents induced in it by the coil. While this effect ers for coupling lines or electrical devices of uncould be eliminated or reduced to a small value equal impedance and has for an object to provide by slitting the sheath, it is desirable because it an impedance transformer which will be substanpermits a much greater rate of change in e than tially independent of the frequency over a Wide would otherwise obtain, and also because it acts 5 range of high frequencies. as an electromagnetic shield for the coil. The

In the preferred embodiment the coupling detapering effect can be controlled by varying the vice takes the form of a tapered transmission diameter of sheath or of coil or both, or by varyline comprising a long cylindrical coil of a large ing the number of turns per unit length of the lo number of turns surrounded by a concentric mecoil. This latter could be accomplished by varytallic sheath of circular cross-section in which ing the pitch of the winding or by using a multithe sheath or the coil is tapered in radius. The layer winding and varying the number of layers. dimensions of the coil and sheath and the rate In the embodiments hereinafter described, single of taper have values dependent upon certain faclayer coils of constant pitch have been used, with 16 tors such as the ratio of the two impedances to either the coil diameter uniform and sheath dil5 be connected by the device and the minimum freameter variable, or vice versa. When a tapered quency to be transmitted as will be explained line is so constructed that its impedance varies hereinafter. In employing such a properly defrom some value, 21, at one end to a different signed device as ahigh frequency transformer bevalue, 22, at the other, it behaves like a transtween a low impedance line and a high impedance former at suiiiciently high frequencies. It is this line one terminal of the low impedance line is property with which We are here concerned. connected to the low impedance end of the coil, Referring to the drawings,

one terminal of the high impedance line is con- Fig. i represents a longitudinal section and Fig.

nected to the high impedance end of the coil, 2 a cross-section of an impedance transformer in o5 and the other two terminals of the two lines are which the coil is of constant radius and the sur- 2,-

connected to opposite ends of the surrounding merounding sheath is tapered;

tallic sheath. The parameters of such a tapered Fig. 3 represents the manner of connecting the line may be readily computed in order that its impedance transformer of Fig. l in a transmisimpedance at each end will be substantially equal sion circuit;

to the impedance of the line connected thereto. Figs. 4 and 5 represent another type of im- 30 The device constructed in this manner will act pedance transformer in which the coll is of taefficiently as a step-up transformer over a wide pered diameter and the surrounding sheath is of range of frequencies and the device is of particuconstant radius;

lar interest in the frequency range above one Fig. 6 is an enlarged longitudinal section of the million cycles per second. tapered line of Fig. 4; 35

A uniform transmission line is characterized by Fig. '7 represents the frequency transmission an impedance, fcharacteristic of the tapered line of Fig. l;

/f Fig. 8 represents the contour of the tapered coil 1=\ line of Fig. 4; and o Fig. 9 represents the contour of the tapered 40 and a, velocity constant, Sheath line 0f Fig, 1 v

1 In determining the parameters of such tapered a :w/ L- line used as an impedance transformer advantage is taken of the observation that when the difwhere L and C are the inductance and capacity ferential equations are expressed in terms of the 45 per unit length of the line. It is possible so to time required fOr a Wave t0 Travel along the construct aline thatLand C vary along its length, line, instead of in terms of the distance, the vegiving thus a tapered line. A flexible and practilocity function drops out. That is, the electrical cal arrangement consists of a solenoid enclosed characteristics of a tapered line depend solely on in a conducting sheath. The ratio of sheath dithe Way in which the impedance varies with the 50 ame-ter to coil diameter at any point then detertime of transmission. Thus all the electrical mines C, while L is determined by this ratio, the characteristics obtainable with tapered lines can number of turns per unit length, and the diambe studied by considering different functions for eter of the coil, at that point. The sheath afthis one parameter. A particular function for 5 fects the inductance, of course, because of curthe impedance having been chosen, the actual 55 shape of the line, for any desired type of construction can be found by direct means.

We begin the mathematical discussion by writing down the fundamental differential equation for a non-dissipative transmission line with a sinusoidal applied voltage of frequency i==current at same point L=inductance per unit length at this point C=capacity per unit length at this point j:/ Expressed in terms of the impedance,

and the velocity,

8:: A l C these equations are:

d'v z ai-wal (l) d 1 "ai Jwza" (2) The time required for a wave to travel to this point from the end where X=0 is X f= 0 Qi 8 from which si i dX- a Since d df d 1 d f=atdt Equations (1) and (2) can be written 'I'he velocity, a, has canceled out, showing that the electrical characteristics of the line depend only on the impedance 2 regarded as a function of the time t.

Diierentiating Equation (2') once with respect to t and using (1') and (2') to eliminate o and 'Ihe rate of taper is fixed by the pure number m. This function leads to solutions of (3) which, for general values of m, contain Bessel functions. When m is an even integer, the solutions are in terms of sines and cosines. In Equation (4), 21

is the value of the line impedance at the low impedance end, say the left-hand end. The physical significance of t1 is explained as follows: Suppose the line were extended to the left until a point was reached where the line impedance was zero. Then t1 is the time required for a wave, started at this end of the line, to reach that part of the line where 2:::1 (the actual extremity of the physical line).

Another type of characteristic is the exponential:

x(t)=z1e" (5) in which the rate of taper is determined by a. 'I'his is a limiting case of (4) obtained by allowing m and t1 to approach infinity in such a way that f1 remains constant,

When the function (4) is used for zu), Equation (3) becomes Then Equation (6) becomes dW clW -1 2 Ls-i-ad--l-[-(E-r) :IW/:0 (7) This is Bessels equation and the general solution for the current in a tapered line having the type of taper given by Equation (4) is thus where @11m-Hol and lf2-ammi are Bessel functions of rst and second kind, respectively, and of order Insertion ganE 20 log). -,db

where I2 is the current in the receiving resistance when an E. M. F. is applied to the other end of the line through the terminal resistance appropriate to that end, and I2' is the current which would iiow in the receiving resistance if the tapered line were removed and the input connected directly to the receiving termination.

The boundary conditions are:

At the low impedance end,

t=0 and E=z|i(0){v(0); at the high impedance end,

V(fz) t t3, say, and i( t2) ,-z,

E is the generator E. M. F.; tz is the time required for a wave to travel the length oi' the line; i(), 11(0) are the current'l and voltage at the low impedance end of the line; i(tz), v(tz) are the current and voltage at the high impedance end; and 21, z2 are the values of the terminal impedances. The constant t2 is determined in terms of zi, zz, t1, and m by Equation (4) wier-1l fl remains constant (so-called exponential line). For the linear line (m=1) Insertion gain=20 log", E 2\/ In the ilrst two cases when f1 is fixed the insertion gain may be plotted as a function of mtl. The ratio .a ti is found from the impedance ratio by Equation (4):

tz=71, for linear line,

Z ;2- l for conical line.

In the exponential case, when at: is fixed the insertion gain may be plotted as a function oi' The parameter atz is determined from the impedance ratio by Equation (5) The quantity t1, in the case of linear and conical lines, or a, in the case oi. exponential lines, is

20 lOgm L wtl'l' D00 1 N fx in which The J- and Y-functions are Bessel functions of the rst and second kind, respectively, and vsatisiy the same recurrence formulae.

For the conical line (m=2) 14,--1 Insertion gain=20 logm-- determined in any specic case by the lowest frequency it is desired to transmit. The cut-oft` frequency of a tapered line is not, in general, sharply defined. Consequently, one should plot the insertion gain for the impedance ratio desired before deciding upon the value of t1 or a).

The above discussion was concerned with the study of possible electrical characteristics of tapered lines without regard to the mechanical For the exponential line (m and t1= y2=a Insertion gain= loglu cos 2 graff-2 -wtl sin 2 t1 fl (u) construction. We shall now show how to determine the shape, size, and number of turns of wire for two particular kinds of mechanical construction, first, where a tapered sheath and a constant radius coil are employed and, second, where a tapered coil and a constant radius sheath are employed.

I-Development of equations for transformer with. tapered sheath First, let the tapered line be made up of a long cylindrical coil of circular cross-section having a constant radius ro and N turns per unit length,

surrounded by a shield with a variable radius r. The inductance per unit length is:

where c=3 101 centimeters per second, the velocity of light.

The factor in the inductance formula takes account of the "short-circuiting effect of the sheath which permits taper in the inductance. The impedance and velocity are given by and let the subscripts 1, 2 reier to the low and high impedance ends of the tapered line, respectively. Then Suppose We consider a perfectly general variation of z with respect to t and write it z=z1f(t) Later, the specic forms of f(t) given by Equations (4) and (5) will be taken up. Diierentiating (22) we have dz 'd-t-Ilf (t) Where f'm-f f'f But Langres dt dydXdt and using (21) we get Now g dy

is found from (19):

dx 1 l-e-U-i-ye This is substituted into (23), Ygiving the differential equation The appearance of a function of t in the integrand may be confusing, since integration is to be performed with respect to y. 'I'he way this is to be handled will be made clear in the specic cases now to be discussed.

Let the function f(t) be that determined by Equation (4):

f(f)=(1 2)" Differentiating lll-l what) But l 1+i-7mm" and from (19) 1 f(f)= -y el?) .Y1 1 8 Hence we have m-l gg y(1-e) tl 1(lge-Ul) The possibility of expressing f(t) explicitly in terms of f(t) was necessary to the solution of the present problem, the determination of a usable relation between X and y to arrive at the shape of the tapered sheath.

Finally, substituting this expression for f'(t) into (24) we get the desired equation This integral of Equation (25) can be expressed in series form, the rst few terms of which will be given for certain values of m. These give suflcient accuracy for practical design purposes. If greater accuracy is desired, more terms must be computed. But it should be noted that whenever a series is diillcult to compute one can always resort to graphical integration.

For the linear line (m=1, (25) becomes For the exponential line or expressed in the series form au 210gyl 96 -l-V II-Development of equations ,for transformer with tapered coil r :radius of coil Tozradius of sheath N=number turns per unit length r1=radius of coil at low impedance end r2=radius of coil at high impedance end y=2 10g fram which wat TIT-1 :a Mg 2,7.

t* (1 0-"luf," and nnauy 2Xm Sat1 which corresponds to Equation (25).

The equations corresponding to (26), (27) and (28) are then:

For the linear line:

It will now be shown how the design is to be carried out for the case where the coil is tapered and the sheath has a constant radius for which the equations are given in Section II. We shall assume a given problem in which the lowest frequency fc which the tapered line is required to pass is 5.2 X106 cycles per second, that the impedance a1 at the small end of the tapered line should be 70 ohms and the impedance z2 at the large end of the tapered line should be '700 ohms.

One must first decide upon the desired taper oi the sheath whether it should be linear, conical or exponential. From the standpoint of electrical characteristics the exponential taper is preferable and will be chosen for the following discussion.

Certain preliminary steps are involved before one can directly nd the parameters of the tapered line for the given values of zi, zz and fc.

Assuming we desire the exponential type of taper we must rst find the value of ata, in order that this value may be used in Equation (12) for the insertion gain of the device.

From Equation It is now necessary to consider the insertion gain characteristic of the exponential tapered line and for this purpose a curve should be plotted with various assigned values of at: =g =10g (40) is 2.43 for our speciic example. Designate this value of Up to the present point the calculations apply to either a tapered sheath or a tapered coil. Ii.' we wish a tapered coil and uniform sheath we have by denition from Equation (30) the following expressions for y1 (the value of y at the low impedance end of the line) and 'J2 (the value of y at the high impedance end .of the line):

Where To is the radius of the sheath, r1 theA radius of the small end of the coil and r2 the radius of the large end of the coil. As an expression involving N, the number of turns per centimeter of the winding surrounded by the sheath, we also have by definition from Equation (29) One is free to specify any one of the quantities y1, y2 or au, after which the others may be uniquely determined. But it is preferable to assign a denite value to yz for it can be shown from Equation (31) that the largest value which can be used for l' is 2.06 and that for a larger value of the impedance of the tapered line will not vary monotonically from one end to the other. Furthermore if we took the shape of the coil form would have an ininite slope at the high impedance end which would be undesirable in practice. Therefore we have the requirement that must be less than 2.06 and since Q .Y2-2 10E r2 this means that y2 should be less than 1.44 in the case of the tapered coil-uniform sheath line. At the same time it must be noted that the smaller y2 is chosen the smaller will be y1 and if y1 is made very small the separation between the sheath and the coil at the low impedance end will also be very small. A study of these factors indicates that y2==1 is a reasonable value.

Assuming this value of 112:1, we now solve for the values of y1 and ao. The value of y1 is to be obtained from Equation (34) letting 2:22 and 1:11a in that equation from which This is a transcendental equation in y1 and its exact solution cannot be obtained algebraically. Although a graphical solution is generally advisable, we can use the first few terms in the power series for an approximate formula for y1 in terms of y2 and Z2 providing the radical 1/yze "(1e-U2) is equal to 0.483. Thus, from the above equation 1li-10.050 Nw from Equation (33) mung 2:22 and 11:11:

9X 1020 10ng X 0.483

We can now choose N and ro in any way as long as their product equals '7.71. If, for example, we make ro=1 inch, then N becomes 7.71 turns per inch, a reasonable value.

'aliases Y lhelengthLofthe-taperedcoilisthatvalue 'or x where y=m and hence rmmnquauon (so) after aiviainqhothmiedery a. `y 2sy v 1 L-F-:Dogjmoe-ym-mw-m]- p es.: cm.=25.7 inches The value oi' X corresponding to each value of y chosen between w=0.050 and m=1 can be computed in the same way. l'br example, ten values ci' y intermediate thesetwo values may be chosen.

The shape oi' the form which supports the tapered coil may be found by plotting elk the value ot being computed for each chosen value of y by v the Formula (30) For these assigned values of y between y1=0.050 and m=1 we can ilnd not only the ratio but also .the distance that ratio holds from the small impedance end ot the line by means of Equation (39) and hence can plot t againstas has been done for our example in Fig. 81'` f-'I'he` curve of Fig. 8, therefore. sives the coil shape for our' specic example.

'I'he radii of the two endsof the 'coil canfrbe readily determined since rq=1 inch and since rr is the value oi' r where `11=y1=0.050 and rz is the value of r where y=yn= 1. That is That is, for our typical example the tapered `ell-0.606 inch i coil ywill be 25.7 inches long, will have a radius of 0.975 inch at its low impedance end, will have a radius of 0.606 inch atits high impedance end and will have the Vcontour prescribed by the curve of Fig. 8. This coil will, of course, be surrounded by a sheath ofconstant radius of 1 inch.

I'I'his construction, it will be recalled, is for coupling a line of 70 ohm impedance to a line of 700 ohm impedance with a lower cut-ofi frequency of 52x10 cyclesper second, and as previously stated thel amount of vinsertion gain in decibels realized by the employment of this device is shown by' Fig. 7.

rv.'cazenlmmm fer meme we involving tapered sheath It 'wm now be shown how the designis to be 4 carried out for the case where the sheath is tapered and the coil is of constant radius, the equations for which are given in Section I. We

shall assume as in the other example in Section against will be that of Fig. 7 and we will utilize the same cut-oi! point where @was from which a=26.9X10 as in previous example.

Again let yz=1 as in previous case, although it may be pointed out that the restriction that 1l: for the tapered coil line should be less than 2.06 does not apply to the tapered sheath construction. We have by deinitlon from Equation (17) where ri is the radius of small impedance end of sheath; ra is radius of large impedance end of sheath; ro is radius of coil; and N is the number of turnsoi' coil per unit length.

Equation` (i9) gives an expression for m in terms ofk the knownquantities ya, zi and zz or and as in the case of the tapered coil we will use the nrst i'ew terms of the power series of eand obtain an approximate solution for y1, namely although a graphical solution of Equation (19) may be employed if desired. n

Now for 112:1 and we obtain from the above formula Also from Equation (18) we obtain from which If we take ro (the radius of the coil) as equal to 1 inch then N=4.68, that is, the coil will have 4.68 turns per inch.

For the length of the coil we have from Equation (28) (where L is the value of X where 1l=ilz after dividing both sides of the equation by gg B0 and by substituting the known values.

L=95.5 cm.=37.6 inches.

The shape of the sheath is computed by solving Equation (28) for X for several values of y between 111:0.081 and :12:1 and then calculating the ratio for each of these values of y by the Formula (17) Finally a curve is plotted of Versus to obtain the shape of the sheath and such a curve is shown in Fig. 10.

If we wish the values of r1, the radius of the sheath at the low impedance end, and n, the radius of the sheath at the high impedance end, we have from Equation (17), knowing that ro=1 inch,

2=e 2 =1.041 inches and EL .1. n=e2=e2=1.649 inches Therefore, when employing a tapered sheath construction for coupling a 70 ohm impedance line to a 700 ohm impedanceline and with a lower cut-off frequency of 5.2 s cycles per second, the coil will have a radius of 1 inch. will be 37.6 inches long, will have 4.68 turns per in'ch and this coil will be surrounded by a sheath having the shape shown by the curve of Fig. l() with a radius of 1.041 inches at the low in ipedance end and a radius of 1.649 inches at v the highimpedance end.

(We will now refer more in detail to the tapered inch. Surrounding the winding Il is a circumferentially complete cylindrical sheath I3 of conducting material such as copper, the shape of the sheath being in conformity with the curve of Fig. 9 previously described. The sheath Il may be suitably supported about the coil II by means of spaced washers I4 of insulating material. This tapered line oi Fig. 1 is shown in cross-section in Fig. 2.

It has been assumed that the low impedance end of the tapered line is to be connected to a line having an impedance of 70 ohms for the irequency range to be transmitted while the high impedance end is to be connected to a line having an impedance of 700 ohms. The manner of connection is shown in Fig. 3 where line I5 having an impedance of 70 ohms has one terminal connected to the terminal I6 of coil II and has its other terminal connected to the surrounding sheath I 3, while line I8 having an impedance of 700 ohms has one terminal connected to terminal I1 of coil Il and has its other terminal connected to sheath I3. 'I'he insertion gain achieved by the use of this impedance transformer for the transmission of current from the frequency source I9 to line I8 is slightly more than 4.5 decibels as shown by Fig. 7 and this substantially constant insertion gain applies for a frequency range of ten to one or greater. y

The other type /of tapered line which has been computed has a tapered coil with a surrounding sheath of constant radius as described in Sections II and III. Such a tapered line is shown in Fig. 4 where the winding 20 is wound on a tapered cylindrical form 2| of insulating material tapered in accordance with the curve of Fig. 8 for the example specifically calculated in order to give the proper taper to the winding. 'I'he winding as shown more clearly in Fig. 6 may comprise ribbon shaped wire fitting into a helical groove in the external surface of the cylinder Z'I. Surrounding the winding 20 is a concentric sheath 23 of conducting material such as copper suitably supported by one or more intermediate insulating washers 24. The core 2l at the high impedance end of the winding may have a collar 25 which supports the sheath at that end while at the low impedance end oi the transformer the sheath 23 may be in contact with core 2I at a point beyond the termination of the winding 20. The tapered line of Fig. 4 may be employed in the transmission circuit of Fig. 3 in the same manner as the tapered line of Fig. 1, the low impedance end oi' winding 20 being connected to line I5 and the high impedance end of winding 20 being connected to line I8.

Although certain specific embodiments of this invention have been described for illustrative purposes it is to be understood that the invention is not hunted thereto as other embodiments can be readily realized commensurate with the scope of the appended claims.

What is claimed is:

l. In combination, two circuits each having a fixed, predetermined impedance, and means connecting said circuits in energy transfer relation comprising an impedance transformer, said impedance transformer comprising two elements. first. a long cylindrical coil and. second. a concentric metallic sheath surrounding said coil, at least one of said elements being tapered in radius'over substantially its entire length. the input and output impedances of said transformer being equal to the respective impedances of said two circuits.

2. In combination, two circuits each having a fixed, predetermined impedance, and means connecting said circuits in energy transfer relation comprising an impedance transformer, said impedance transformer comprising two elements, rst, a long cylindrical coil, and second, a concentric metallic sheath surrounding said coil, one oi' said elements being tapered in radius over substantially its entire length, the other of said elements being of substantially constant radius over its entire length, the input and output impedances of said transformer being equal to the respective impedances of said two circuits.

3. In combination, two circuits each having a fixed, predetermined impedance, and means connecting said circuits in energy transferl relation comprising an impedance transformer, said impedance transformer comprising a long single layer solenoidal winding of substantially constant radius throughout its length and a metallic sheath of substantially circular cross-section surrounding said winding, said sheath being tapered in radius over substantially its entire length, the input and output impedances of said transformer being equal to the respective impedances of said two circuits.

4. In combination, two circuits each having a fixed, predetermined impedance, and means connecting said circuits in energy transfer relation comprising an impedance transformer, said impedance transformer comprising a long single layer solenoidal winding and a metallic sheath of substantially circular cross-section surrounding said winding, said sheath being of substantially constant radius throughout its length, the radius of said winding being tapered over substantially its entire length, the input and output imped` ances of said transformer being equal to the respective impedances of said two circuits.

5. An impedance transformer comprising a long cylindrical coil and aconcentric metallic sheath surrounding said coil, said coil being tapered in radius over substantially its entire length, said sheath being of substantially constant radius, the ratio of sheath radius to coil radius at the high impedance end of said transformer being less than 2.06. 4

6. A radio frequency impedance matching device Whose terminals are adapted to face different impedances comprising two elements, rst, a solenoidal winding, and second, a conducting sheath enclosing said winding, one set of terminals for said device comprising one end of said winding and the adjacent end of said sheath, the other set of terminals for said device comprising the other end of said Winding and the adjacent end of said sheath, said elements having such parameters that the impedance of said device tapers substantially exponentially as the time of Wave transmission along its length, whereby its terminals present impedances having values which are substantially equal to the impedances to be faced.

7. A radio frequency impedance matching device whose terminals are adapted to face different impedances comprising two elements, first, a long cylindrical shaped winding, and second, a conducting sheath enclosing said winding, the radius of at least one of said elements varying substantially continuously along its length, one set of terminals for said device comprising one end of said winding and the adjacent end of said sheath, the other set of terminals for said device comprising the other end of said winding and the adjacent end of said sheath, said elements having such parameters that the impedance of said device tapers approximately exponentially as the time of wave transmission along its length whereby its terminals present impedances having values which are substantially equal to the impedances to be faced.

8. A radio frequency impedance matching device in accordance with claim 7 in which said winding is tapered in radius along its length and said sheath is of substantially constant radius.

9. A radio frequency impedance matching device in accordance with claim 7 in which said sheath is tapered substantially continuously alongV its length and said winding is of substantially constant radius.

10. In combination, a pair of radio frequency electrical circuits having different impedances, an impedance matching device whose terminals are adapted to connect together said pair of circuits, each set of terminals of said device presenting an impedance substantially equal to the impedance of the electric circuit which it faces, said device comprising a long cylindrical coil and a concentric conducting sheath surrounding said coil, one set of terminals of said device comprising one end of said Winding and the adjacent end of said sheath, the other set of terminals of said device comprising the other end of said winding and the adjacent end of said sheath, said device having an impedance which tapers continuously along its length substantially in accordance with Equation (4).

11. An impedance transformer comprising a long cylindrical shaped winding and a concentric metallic tapered sheath surrounding said winding in which the parameters of the transformer are substantially dened by the formula l +eV-1 1 m-ldy wlan-(5%)?? tryo-WNW where X=axial length of winding ro=radius of winding N=number of turns of winding per unit length r y-2 log t--c r=radius of sheath at any point along its length r1=radius of sheath at low impedance end t1 is the value obtained from the expression l [film-1 12. An impedance transformer comprising a long cylindrical shaped winding whose radius is tapered along its axis and a concentric metallic sheath surrounding said winding in which the parameters of the transformer are substantially defined by the formula l -y t evl X: an l i m-idy N=number of turns of coil per unit length t1=value of expression y=2 log L;

r=radus of winding at any point along its axis m=a constant preferably one of the following values: 1, 2, infinity e=base of natural logarithms 13. In combination, two electrical circuits having different impedances and an impedance audace matching device interconnecting them for the transfer of radio frequency waves occupying a wide range of radio frequencies such that the highest frequency of said range is at least several times the lowest frequency, said device comprising a long cylindrical coil and a concentric metallic sheath surrounding said coil, at least one of said elements being tapered in radius over substantially its entire length in such manner that the impedance varies with the time of transmission of said waves in accordance with Equation (4).

14. In combination, two electrical circuits having different impedances and an impedance matching transmission line interconnecting them for the transfer of radio frequency waves comprising a long cylindrical coil and a concentric conducting sheath surrounding said coil, the impedance of said line varying gradually along said line with the time of transmission in such manner that the insertion gain is substantially uniin which. said line is tapered in impedance in accordance with Equation (4), where m lies between 2 and infinity.

EDWARD L. NORTON. RONALD F'. WICK. 

